The chain view computes u = 3x + 1 and y = u squared. Its derivative is the product 3 times 2u. The two-path view computes u = x squared, v = x + 1, and z = uv. Its derivative adds two contributions: v times 2x, and u times 1. The input slider updates the local derivatives and their total. The animation traces first-order changes; the readout gives the same information without animation.
Along each route, multiply the local derivatives. When x reaches the output by several routes, add their contributions. The animation uses dx for an infinitesimal input change. For a finite change, derivative × change is a first-order approximation.